A sinusoid (sine curve) rises from a minimum value of y=-8 at x=0 to a maximum value of y=40 at x=50. Write a sine function for this curve
The function has a mean value of (40-8)/2 = 16 (which will have to be added as a constant term), and it attains the mean value when x = 25. The amplitude is 40-16 - 24. Since is goes from min to max when x changes by 50, the period is 100
F(x) = 16 + 24 sin[2 pi (x-25)/100]
= 16 + 24 sin[pi(x-25)/50]
To write a sine function for this curve, we need to know the amplitude, period, and phase shift of the function.
1. Amplitude: The amplitude is the distance from the midline of the sinusoid to the maximum or minimum value. In this case, the minimum value is y = -8, and the maximum value is y = 40. Thus, the amplitude is half the difference between the maximum and minimum values: (40 - (-8)) / 2 = 48 / 2 = 24.
2. Period: The period is the distance it takes for the sinusoid to complete one full cycle. In this case, the sinusoid is at its starting point (the minimum value) at x = 0 and reaches its maximum value at x = 50. Therefore, the period is 50 - 0 = 50.
3. Phase Shift: The phase shift is the horizontal displacement of the sine curve. In this case, the sinusoid is at its starting point (minimum value) at x = 0. Hence, there is no horizontal displacement or phase shift.
Putting it all together, we can write the sine function for this curve as:
y = A * sin(Bx) + C
- A is the amplitude: A = 24
- B is the frequency: B = (2π) / period = (2π) / 50
- C is the vertical shift: C = (maximum value + minimum value) / 2 = (40 + (-8)) / 2 = 16 / 2 = 8
Therefore, the sine function for this curve is:
y = 24 * sin((2π/50)x) + 8